Intensity-modulated radiotherapy (IMRT) has revolutionized the treatment of cancers in the last decade, since it can tightly conform and escalate radiation dose to a tumor while simultaneously protecting nearby radiation- sensitive normal tissues, resulting in better local control and fewer post-treatment complications than previous techniques. However, the process of obtaining a clinically acceptable IMRT plan for a difficult site is still extremely slow, requiring many hours of a busy expert's time in a manual trial-and-error loop of parameter adjustment. The goal of this project is to drastically reduce the amount of time to obtain a clinically acceptable IMRT plan using a new automated method that directly applies constrained optimization in a computationally tractable and clinically meaningful way. The hypothesis is that clinical treatment planning times using this technique will be reduced from several hours to a matter of minutes. The new approach, called ROCO (Reduced-Order Constrained Optimization) translates well-established concepts from optimization and machine learning theory to the novel application of IMRT planning, exploiting the speed and ease of unconstrained optimizations and introducing a dimensionality reduction step that makes true constrained optimization tractable. The Specific Aims of the proposal are to (1) apply Reduced-Order Con- strained Optimization to IMRT planning for non-small cell lung cancers and nasopharynx cancers, where the planning process is highly time-consuming; (2) develop and extend the Reduced-Order Constrained Optimization paradigm to a promising IMRT variant called Volumetric Modulated Arc Therapy (VMAT) for the prostate site, which is currently nearly clinically intractable to plan; and (3) integrate the new tools into the clinical IMRT planning process at Memorial Sloan-Kettering Cancer Center, using a powered study to verify the hypothesis that the proposed method significantly improves planning speed. The experiments will be designed in consultation with an expert clinical treatment planner and biostatistician, and carefully validated using anonymized data from approximately 50 patients for each site. The main benefit of the proposed approach is to drastically reduce planning times, which is critical if IMRT and VMAT are to reach their full potential in clinical application. In a busy clinic, long planning times place a severe stress on available resources, and can result in treatment delays, acceptance of sub-optimal plans or - in the worst case - errors due to time pressure. In the longer term, the proposed approach will provide deeper insight into the critical elements of the dose optimization problem, significantly reduce the trial-and-error effort characteristic of current IMRT planning, and reduce subjectivity in treatment plan selection.